Integrand size = 24, antiderivative size = 108 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx=4 i a^3 x+\frac {4 i a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^2(c+d x)}{d}-\frac {3 i a^3 \cot ^3(c+d x)}{4 d}+\frac {4 a^3 \log (\sin (c+d x))}{d}-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d} \]
4*I*a^3*x+4*I*a^3*cot(d*x+c)/d+2*a^3*cot(d*x+c)^2/d-3/4*I*a^3*cot(d*x+c)^3 /d+4*a^3*ln(sin(d*x+c))/d-1/4*cot(d*x+c)^4*(a^3+I*a^3*tan(d*x+c))/d
Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx=a^3 \left (\frac {4 i \cot (c+d x)}{d}+\frac {2 \cot ^2(c+d x)}{d}-\frac {i \cot ^3(c+d x)}{d}-\frac {\cot ^4(c+d x)}{4 d}+\frac {4 \log (\tan (c+d x))}{d}-\frac {4 \log (i+\tan (c+d x))}{d}\right ) \]
a^3*(((4*I)*Cot[c + d*x])/d + (2*Cot[c + d*x]^2)/d - (I*Cot[c + d*x]^3)/d - Cot[c + d*x]^4/(4*d) + (4*Log[Tan[c + d*x]])/d - (4*Log[I + Tan[c + d*x] ])/d)
Time = 0.81 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4036, 25, 3042, 4074, 27, 3042, 4012, 3042, 4012, 25, 3042, 4014, 3042, 25, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^3}{\tan (c+d x)^5}dx\) |
| \(\Big \downarrow \) 4036 |
| \(\displaystyle -\frac {1}{4} \int -\cot ^4(c+d x) (i \tan (c+d x) a+a) \left (9 i a^2-7 a^2 \tan (c+d x)\right )dx-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \int \cot ^4(c+d x) (i \tan (c+d x) a+a) \left (9 i a^2-7 a^2 \tan (c+d x)\right )dx-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {(i \tan (c+d x) a+a) \left (9 i a^2-7 a^2 \tan (c+d x)\right )}{\tan (c+d x)^4}dx-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 4074 |
| \(\displaystyle \frac {1}{4} \left (\int -16 \cot ^3(c+d x) \left (i \tan (c+d x) a^3+a^3\right )dx-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-16 \int \cot ^3(c+d x) \left (i \tan (c+d x) a^3+a^3\right )dx-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (-16 \int \frac {i \tan (c+d x) a^3+a^3}{\tan (c+d x)^3}dx-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (-\frac {a^3 \cot ^2(c+d x)}{2 d}+\int \cot ^2(c+d x) \left (i a^3-a^3 \tan (c+d x)\right )dx\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (-\frac {a^3 \cot ^2(c+d x)}{2 d}+\int \frac {i a^3-a^3 \tan (c+d x)}{\tan (c+d x)^2}dx\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (\int -\cot (c+d x) \left (i \tan (c+d x) a^3+a^3\right )dx-\frac {a^3 \cot ^2(c+d x)}{2 d}-\frac {i a^3 \cot (c+d x)}{d}\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (-\int \cot (c+d x) \left (i \tan (c+d x) a^3+a^3\right )dx-\frac {a^3 \cot ^2(c+d x)}{2 d}-\frac {i a^3 \cot (c+d x)}{d}\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (-\int \frac {i \tan (c+d x) a^3+a^3}{\tan (c+d x)}dx-\frac {a^3 \cot ^2(c+d x)}{2 d}-\frac {i a^3 \cot (c+d x)}{d}\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (-a^3 \int \cot (c+d x)dx-\frac {a^3 \cot ^2(c+d x)}{2 d}-\frac {i a^3 \cot (c+d x)}{d}-i a^3 x\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (-a^3 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a^3 \cot ^2(c+d x)}{2 d}-\frac {i a^3 \cot (c+d x)}{d}-i a^3 x\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (-16 \left (a^3 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a^3 \cot ^2(c+d x)}{2 d}-\frac {i a^3 \cot (c+d x)}{d}-i a^3 x\right )-\frac {3 i a^3 \cot ^3(c+d x)}{d}\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {1}{4} \left (-\frac {3 i a^3 \cot ^3(c+d x)}{d}-16 \left (-\frac {a^3 \cot ^2(c+d x)}{2 d}-\frac {i a^3 \cot (c+d x)}{d}-\frac {a^3 \log (-\sin (c+d x))}{d}-i a^3 x\right )\right )-\frac {\cot ^4(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{4 d}\) |
(((-3*I)*a^3*Cot[c + d*x]^3)/d - 16*((-I)*a^3*x - (I*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^2)/(2*d) - (a^3*Log[-Sin[c + d*x]])/d))/4 - (Cot[c + d* x]^4*(a^3 + I*a^3*Tan[c + d*x]))/(4*d)
3.1.32.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x] )^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] + Si mp[a/(d*(b*c + a*d)*(n + 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[ e + f*x])^(n + 1)*Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b *c - a*d)*(A*b - a*B)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2 ))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*A*c + b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m , -1] && NeQ[a^2 + b^2, 0]
Time = 0.47 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (-\left (\cot ^{4}\left (d x +c \right )\right )-4 i \left (\cot ^{3}\left (d x +c \right )\right )+16 i d x +8 \left (\cot ^{2}\left (d x +c \right )\right )+16 i \cot \left (d x +c \right )+16 \ln \left (\tan \left (d x +c \right )\right )-8 \ln \left (\sec ^{2}\left (d x +c \right )\right )\right )}{4 d}\) | \(75\) |
| derivativedivides | \(\frac {a^{3} \left (4 i \cot \left (d x +c \right )-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-i \left (\cot ^{3}\left (d x +c \right )\right )+2 \left (\cot ^{2}\left (d x +c \right )\right )-2 \ln \left (\cot ^{2}\left (d x +c \right )+1\right )-4 i \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\) | \(78\) |
| default | \(\frac {a^{3} \left (4 i \cot \left (d x +c \right )-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-i \left (\cot ^{3}\left (d x +c \right )\right )+2 \left (\cot ^{2}\left (d x +c \right )\right )-2 \ln \left (\cot ^{2}\left (d x +c \right )+1\right )-4 i \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )\right )}{d}\) | \(78\) |
| risch | \(-\frac {8 i a^{3} c}{d}-\frac {2 a^{3} \left (12 \,{\mathrm e}^{6 i \left (d x +c \right )}-23 \,{\mathrm e}^{4 i \left (d x +c \right )}+18 \,{\mathrm e}^{2 i \left (d x +c \right )}-5\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(88\) |
| norman | \(\frac {-\frac {a^{3}}{4 d}+\frac {2 a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+4 i a^{3} x \left (\tan ^{4}\left (d x +c \right )\right )-\frac {i a^{3} \tan \left (d x +c \right )}{d}+\frac {4 i a^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{d}}{\tan \left (d x +c \right )^{4}}+\frac {4 a^{3} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {2 a^{3} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(117\) |
1/4*a^3*(-cot(d*x+c)^4-4*I*cot(d*x+c)^3+16*I*d*x+8*cot(d*x+c)^2+16*I*cot(d *x+c)+16*ln(tan(d*x+c))-8*ln(sec(d*x+c)^2))/d
Time = 0.23 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.61 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {2 \, {\left (12 \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 23 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 5 \, a^{3} - 2 \, {\left (a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
-2*(12*a^3*e^(6*I*d*x + 6*I*c) - 23*a^3*e^(4*I*d*x + 4*I*c) + 18*a^3*e^(2* I*d*x + 2*I*c) - 5*a^3 - 2*(a^3*e^(8*I*d*x + 8*I*c) - 4*a^3*e^(6*I*d*x + 6 *I*c) + 6*a^3*e^(4*I*d*x + 4*I*c) - 4*a^3*e^(2*I*d*x + 2*I*c) + a^3)*log(e ^(2*I*d*x + 2*I*c) - 1))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)
Time = 0.34 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.53 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {4 a^{3} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 24 a^{3} e^{6 i c} e^{6 i d x} + 46 a^{3} e^{4 i c} e^{4 i d x} - 36 a^{3} e^{2 i c} e^{2 i d x} + 10 a^{3}}{d e^{8 i c} e^{8 i d x} - 4 d e^{6 i c} e^{6 i d x} + 6 d e^{4 i c} e^{4 i d x} - 4 d e^{2 i c} e^{2 i d x} + d} \]
4*a**3*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-24*a**3*exp(6*I*c)*exp(6*I*d* x) + 46*a**3*exp(4*I*c)*exp(4*I*d*x) - 36*a**3*exp(2*I*c)*exp(2*I*d*x) + 1 0*a**3)/(d*exp(8*I*c)*exp(8*I*d*x) - 4*d*exp(6*I*c)*exp(6*I*d*x) + 6*d*exp (4*I*c)*exp(4*I*d*x) - 4*d*exp(2*I*c)*exp(2*I*d*x) + d)
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.87 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {-16 i \, {\left (d x + c\right )} a^{3} + 8 \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 16 \, a^{3} \log \left (\tan \left (d x + c\right )\right ) + \frac {-16 i \, a^{3} \tan \left (d x + c\right )^{3} - 8 \, a^{3} \tan \left (d x + c\right )^{2} + 4 i \, a^{3} \tan \left (d x + c\right ) + a^{3}}{\tan \left (d x + c\right )^{4}}}{4 \, d} \]
-1/4*(-16*I*(d*x + c)*a^3 + 8*a^3*log(tan(d*x + c)^2 + 1) - 16*a^3*log(tan (d*x + c)) + (-16*I*a^3*tan(d*x + c)^3 - 8*a^3*tan(d*x + c)^2 + 4*I*a^3*ta n(d*x + c) + a^3)/tan(d*x + c)^4)/d
Time = 1.32 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.67 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx=-\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 108 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1536 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 768 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 456 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {1600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 456 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 108 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
-1/192*(3*a^3*tan(1/2*d*x + 1/2*c)^4 - 24*I*a^3*tan(1/2*d*x + 1/2*c)^3 - 1 08*a^3*tan(1/2*d*x + 1/2*c)^2 + 1536*a^3*log(tan(1/2*d*x + 1/2*c) + I) - 7 68*a^3*log(tan(1/2*d*x + 1/2*c)) + 456*I*a^3*tan(1/2*d*x + 1/2*c) + (1600* a^3*tan(1/2*d*x + 1/2*c)^4 - 456*I*a^3*tan(1/2*d*x + 1/2*c)^3 - 108*a^3*ta n(1/2*d*x + 1/2*c)^2 + 24*I*a^3*tan(1/2*d*x + 1/2*c) + 3*a^3)/tan(1/2*d*x + 1/2*c)^4)/d
Time = 5.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.74 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}}{d}-\frac {-a^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\,4{}\mathrm {i}-2\,a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2+a^3\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}+\frac {a^3}{4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4} \]